Spin excitations in magnetic structures of different dimensions - - PowerPoint PPT Presentation

Spin excitations in magnetic structures

• f different dimensions

Wulf Wulfhekel Physikalisches Institut, Universität Karlsruhe (TH) Wolfgang Gaede Str. 1, D-76131 Karlsruhe

• 0. Overview

Chapters of spin excitation

• 1. Why are excitations of any importance?
• 2. Excitations of ferromagnets in the Heisenberg model
• 3. Excitations of antiferromagnets in the Heisenberg model
• 4. Spin waves in bulk, thin films and stripes
• 5. Itinerant magnetism
• 6. Experimental techniques to study excitations

questions

• 1. Why are magnetic excitations of any importance?

Magnetic data storage

Write poles and GMR sensors

• 1. Why are magnetic excitations of any importance?

Reading (and writing) data from a disk

• 1. Why are magnetic excitations of any importance?

Typical data speed: 120MB/sec = 1GHz

Superparamagnetism A single domain particle with e.g. uniaxial magnetic anisotropy due to magnetocrystalline anisotropy

• r a elongated shape (shape anisotropy) has two

states with minimal energy. In case the energy barrier given by the anisotropy cannot be overcome thermally within a certain time, the magnetic moment is stable. In case the barrier can be overcome, the magnetic moment flips randomly between the states and the particle becomes superparamagnetic.

• 1. Why are magnetic excitations of any importance?

The magnetic moment of a bound electron  l=I  A=−e r

2=−e

2m mr

2=−e

2m ℏ l =−B l B= e ℏ 2m=9.27×10

−24 J /T

Magnetic moment of ring current (orbital moment) Bohr magneton Magnetic moment of spin (spin moment)  S=−B g  s Landé factor of the electron g=2.0023≈2 In bulk spin moment usually dominates s≫l Attention: The magnetic moment behaves like an angular moment (precession).

• 1. Why are magnetic excitations of any importance?

µMAG standard problem #4

NIST, Maryland (VA) USA, M. Donahue et al. http://www.ctcms.nist.gov/~rdm/mumag.org.html

Ground state of magnetic particle is single domain. Dynamic of magnetization reversal

• 1. Why are magnetic excitations of any importance?

µ0H=25mT

φ

=170 °

• 1

+1

my

During switcheing the partcile is not Single domain anymore. The magnetistatic energy is converted to spin waves.

<mx> <my> <mz>

Magnetization dynamics

• 1. Why are magnetic excitations of any importance?

• 0. Overview

Chapters of spin excitation

• 1. Why are excitations of any importance?
• 2. Excitations of ferromagnets in the Heisenberg model
• 3. Excitations of antiferromagnets in the Heisenberg model
• 4. Spin waves in bulk, thin films and stripes
• 5. Itinerant magnetism
• 6. Experimental techniques to study excitations

questions

• 2. Excitations in ferromagnets in the Heisenberg model

Direct exchange interaction between two electrons Quantum mechanical system with two electrons : total wave function must be antisymmetric under exchange of the two electrons, as electrons are fermions. 1,2=−2,1 Wave function of electron is a product of spatial and spin part: 1=r1× 1 For antiparallel spins (singlet): (↑↓ - ↓↑) antisymmetric For parallel spins (triplet) : = ↑↑, (↑↓ + ↓↑), ↓↓ symmetric 1,2 1

2

 1,2= 1

2

→ Spatial part of wave function has opposite symmetry to spin part

Direct exchange interaction between two electrons r1,r2= 1

2 ar1br2ar2br1

→ Coulomb repulsion is lower for antisymmetric spatial wave function and thus its energy Is lower than that of the symmetrical spatial wave function r1,r2= 1

2 ar1br2−ar2br1

symmetric for singlet antisymmetric for triplet For the antisymmetric wave function : r1, r2=−r2, r1 In case r1=r2 follows : r ,r=0 Exchange interaction between two spins: difference of the coulomb energy due to symmetry E S−ET=2∫a

*r1b *r2

e

2

40∣r1−r2∣ar2br1dr1dr2

• 2. Excitations in ferromagnets in the Heisenberg model

Direct exchange between localized electrons J = ES−ET 2 , Eex=−2J  S1  S 2 J>0 : parallel spins are favoured (ferromagnetic coupling) J<0: antiparallel spins are favoured (antiferromagnetic coupling) Heisenberg model for N spins: Nearest neighbor Heisenberg model: E=− ∑

i,j=1 N

J ij  S i  S j E=−∑

i,j NN

J  S i  S j As electrons are assumed as localized, wave functions decay quickly and mainly nearest neighbors contribute to exchange.

• 2. Excitations in ferromagnets in the Heisenberg model

Ferromagnetism J>0 Spins align in parallel at T=0 Elements : Fe, Co, Ni, Gd … Oxides : Fe2O3, CrO … Semiconductors : GaMnAs, EuS ... Above Curie temperature Tc, they become paramagnetic. Fe 1043K EuS 16.5K Co 1394K GaMnAs

• ca. 180K

Ni 631K Gd 289K T C≈ 2 z J ex J J 1 3k B z nearest neighbors

• 2. Excitations in ferromagnets in the Heisenberg model

Solving the excitation spectrum Quantum mechanically exact solution is extremely hard if not impossible. N coupled atoms of spin S have a (2S+1)N dimensional Hilbert space. Example: A 3x3x3 Fe cluster (S=2) has 527 = 745.058.059.623.827.125 states. Let us try in a 1D chain of atoms (S=1/2) with only nearest neighbor interactions:

H =−2 J ∑

j=i+1

 S i  S j=−2J∑

i

S i

z Si1 z

 1 2 S i

+ Si1

• Si
• S i1

+ 

S

+=S xiS y

S

• =S x−iS y

S

+ |1

2 >=0 S

• |− 1

2 >=0 S

+ |− 1

2 >=| 1 2 > S

• | 1

2 >=|−1 2 >

• 2. Excitations in ferromagnets in the Heisenberg model

Solving the excitation spectrum H | j >=2−NJS

22JS 2| j >−SJ | j1 >−SJ | j−1>

j The single flipped spin is no eigenstate of the Hamiltonian! Naïve try for an excited state: .... .... | j > = j-th spin is flipped For the ground state (say Si=+S) : H | >=−2NJS

2|>

• 2. Excitations in ferromagnets in the Heisenberg model

Solving the excitation spectrum for a 1D ferromagnetic chain Solution: excited states are described by the ground state plus small excitations named magnons (quasiparticles) that do not interact. Shortcomings: in reality, magnons do interact. H =−2 J ∑

j=i+1

 S i  S j Semiclassical ansatz: S j

z≈S

S j

x=A e i qja− t

S j

y=B e iqja−t

Solution: ℏ =4JS1−cos qa See blackboard! 8JS  a

• 2. Excitations in ferromagnets in the Heisenberg model

Magnons or spin waves Excitation of spin 1

• 2. Excitations in ferromagnets in the Heisenberg model

q=2 

Magnons

Lindis Pass, NZ

• 2. Excitations the ferromagnets in the Heisenberg model

Magnons in ferromagnets

from Blundell

fcc Co

• 2. Excitations in ferromagnets in the Heisenberg model

Dispersion : ℏ =4JS1−cos qa≈2JSa

2 q 2=D q 2

D is called spin wave stiffness and behaves like an inverse mass Fe Co Ni D [meVÅ2] 281 500 364

• 0. Overview

Chapters of spin excitation

• 1. Why are excitations of any importance?
• 2. Excitations of ferromagnets in the Heisenberg model
• 3. Excitations of antiferromagnets in the Heisenberg model
• 4. Spin waves in bulk, thin films and stripes
• 5. Itinerant magnetism
• 6. Experimental techniques to study excitations

questions

Antiferromagnetism J<0 Spins align antiparallel at T=0 Elements : Mn, Cr … Oxides : FeO, NiO … Semiconductors : URu2Si2 … Salts : MnF2 ... Above Néel temperature TN, they become paramagnetic. Cr 297K FeO 198K NiO 525K

• 3. Excitations in antiferromagnets in the Heisenberg model

Antiferromagnetic configurations Depending on the crystal structure, many different antiferromagnetic configurations may exist.

• 3. Excitations in antiferromagnets in the Heisenberg model

Magnon dispersion of antiferromagnets

• 3. Excitations in antiferromagnets in the Heisenberg model

Solution: two ferromagnetic sublattices that couple antiferromagnetically. H =−2 J ∑

j=i+1

 Si  S j, J 0 Ansatz: S 2p

z =S ,S 2p1 z

=−S Solution: ℏ =−4JS∣sinqa ∣ See blackboard! −4JS  a

Magnones in antiferromagnets

from Kittel

RbMnF3

• 3. Excitations in antiferromagnets in the Heisenberg model

Dispersion : ℏ =−4JSsin qa≈−4JSa q=v q v is called spin wave velocity; magnons behave like massless objects

Spinons

• 3. Excitations in antiferromagnets in the Heisenberg model

.... .... Ground state .... .... Excited state ΔS=1 .... .... Excited state ΔS=1/2 Magnon Spinon Magnon= 2 * Spinon KCuF3

From Helmholtz Center Berlin

• 0. Overview

Chapters of spin excitation

• 1. Why are excitations of any importance?
• 2. Excitations of ferromagnets in the Heisenberg model
• 3. Excitations of antiferromagnets in the Heisenberg model
• 4. Spin waves in bulk, thin films and stripes
• 5. Itinerant magnetism
• 6. Experimental techniques to study excitations

questions

Thermodynamics of ferromagnets

• 4. Spin waves in bulk, tin films and stripes

Bloch´s T3/2 law < nq >= 1 e

ℏ q/kT−1

Thermal occupation number of a magnon: Total number of magnons: N = ∫

BZ < nq > dq=∫

• ∞

Dnd  With parabolic dispersion within the BZ and in 3 dimensions: D

1/2

N  T

3/2

D

Thermodynamics of magnets

M S∝∣T C−T∣

β

χ∝∣T−T C∣

−γ

M T=T C∝±∣H∣

1 δ

ξ ∝∣T C−T∣

−ν

T>TC T<TC Phase transition between ferromagnetic (T<Tc) and paramagnetic (T>Tc). Magnetic phase transitions are

• f 2nd order, i.e. M continuously

goes to zero when Tc is approached. Critical exponents describe the properties of the magnet near Tc.

∣T −T C∣



• 4. Spin waves in bulk, tin films and stripes

1−T

3/2

Critical exponents of some model systems 0,705 4,803 1,387 0,365 3d-Heisenberg 0,669 4,810 1,316 0,345 3d-XY 0,630 4,816 1,240 0,325 3d-Ising 1,33 10,6 2,2 0,23 2d-XY 1 15 1,75 0,125 2d-Ising 0,5 3 1 0,5 Landau-Theory ν δ γ β Exponent Ising : spin can only point along +z direction XY : spin lies in the xy-plane Heisenberg : spin can point in any direction in space Landau : classical theory

• 4. Spin waves in bulk, tin films and stripes

The Mermin-Wagner theorem The Mermin-Wagner theorem predicts Tc=0K for three dimensional spins in two dimensions that interact via the exchange interaction. A Kosterlitz-Thouless phase transition (self similar vortex state) is predicted for T=0. Spin waves

• f parabolic dispersion

in 2 dimensions:

• 4. Spin waves in bulk, tin films and stripes

D=const

2D Heisenberg - model 2 atomic layers of Fe/W(100) Two easy direction in the film plane, hard axis normal to the plane Expected ordering temperature 0K,

• bserved 207K

HJ Elmers, J. Appl. Phys. (1996)

• 4. Spin waves in bulk, tin films and stripes

Beyond exchange interaction

• 4. Spin waves in bulk, tin films and stripes

The magnetic moments of a ferromagnet feel other forces than only the exchange. Zeeman energy density : Dipolar energy density : Anisotropy energy density : H =−0 g B  H ext  S H = f  S  e.g.: H =K cos

2

S , z=K S z

2

H =∫

V

0 M 2  m r  ∇ ´  m r ´ r ´− r 4∣ r− r ´∣

3

d  r ´ These can be written as an effective field favouring a certain direction in the ground state. A magnon, in which the spin deviates from this ground state, will cost additional energy. For the magnon at q=0, i.e. the coherent rotation, the frequency is then given by the Larmor frequency of the spin in the effective field: ℏ q=0=0 g B H eff A magnon gap evolves!

Limits of the Mermin-Wagner theorem Even slightest anisotropies lead to break down of Mermin-Wagner theorem (magnon gap). A magnetization for T>0 results. When film thickness increases, the ordering temperature of the 2D-system quickly approaches that of the 3D system.

• 4. Spin waves in bulk, tin films and stripes

1 ML Fe/W(110): 2D-Ising Uniaxial magnetic anisotropy in the film plane results in 2D Ising model Critical exponent: β=0.133 (0.125)

HJ Elmers, Phys. Rev. B (1996)

• 4. Spin waves in bulk, tin films and stripes

Thin film modes inc. dipolar and shape anisotropy

• 4. Spin waves in bulk, tin films and stripes

At large q, spin wave dispersion is dominated by exchange At small q, spin wave dispersion is dominated by dipolar energy (shape anisotropy)

Thin film modes inc. dipolar and shape anisotropy

• 4. Spin waves in bulk, tin films and stripes

B0 : effective field defining ground state Forward volume spin waves Backward volume spin waves Surface spin waves vG=  ∇ k 0

Figure: B. Hillebrands

Thin film modes inc. dipolar and shape anisotropy

• 4. Spin waves in bulk, tin films and stripes

Thin film modes inc. dipolar and shape anisotropy

• 4. Spin waves in bulk, tin films and stripes

Perpendicular to the plane, the modes have high q due to small thickness. Thus the modes are determined by the exchange leading to open boundary conditions.

Figure: B. Hillebrands

Magnon modes in stripes

• 4. Spin waves in bulk, tin films and stripes

In the stripe plane, the modes have small q due to large width (tens to hundreds of nm). Thus the modes are determined by dipolar energy leading to fixed boundary conditions (nodes) to avoid magnetic surface charges.

Figure: B. Hillebrands

The 1D-Ising chain …

N ∞⇒ ΔS ∞

F=UΔE −T SΔS −∞

H =−∑

i=1 N

J  S i

z  

S i1

 z  =−1

2 NJ

1 2 3 4 5 6 7 8 9 10 11 N N+1

…

1 2 3 4 5 6 7 8 9 10 11 N N+1

One domain wall Energy cost: Entropy gain: long Ising chain: ∞ Entropy wins and no ordering occurs.  E= J 4  S=k BlnN 

• 4. Spin waves in bulk, tin films and stripes

1D Ising chain M=0 for H=0 independent of temperature. Experimental realisation by step edge decoration of Cu(111) steps with Co. Co shows magnetization perpendicular to the plane due to surface anisotropy.

• 4. Spin waves in bulk, tin films and stripes

Glauber dynamic Experiment shows remanence in the MOKE loop. Magnetization is only metastable.

J.Shen Phys. Rev. B (1997)

• 4. Spin waves in bulk, tin films and stripes

• 0. Overview

Chapters of spin excitation

• 1. Why are excitations of any importance?
• 2. Excitations of ferromagnets in the Heisenberg model
• 3. Excitations of antiferromagnets in the Heisenberg model
• 4. Spin waves in bulk, thin films and stripes
• 5. Itinerant magnetism
• 6. Experimental techniques to study excitations

questions

• 5. Itinerant magnetism

Slater-Pauling curve

Band structure of fcc Ni Electrons are delocalized and form electron bands. In itinerant ferromagnets, bands are spin split and thus electron occupation for spin up and spin down differ. Difference can be a non integer number resulting in irrational spin moments per atom.

• 5. Itinerant magnetism

Stoner criterion

• 5. Itinerant magnetism

Cost due to kinetic energy: n electrons E kin= E n0 n= g E F 2  E  Ekin= g EF 2  E2 Magnetization: M =Bndown−nup=2B n=B gE F E Potential energy : dF=−M dB=−0 M dM E pot=−1 2 U exg E F E2 Spontaneous magnetization develops for : U ex g EF1

Stoner excitations and magnon life times

• 5. Itinerant magnetism

ℏ  k U ex Excitation of an majority electrons below to a minority electron above the Fermi edge. Excitation has spin 1, a wave vector and an energy, just like a magnon but forms a continuum. Stoner continuum Magnon ℏ  k U ex Magnons and Stoner excitations can couple where they overlap leading to magnon decay into Stoner excitations and thus to short magnon life times (damping).

• 0. Overview

Chapters of spin excitation

• 1. Why are excitations of any importance?
• 2. Excitations of ferromagnets in the Heisenberg model
• 3. Excitations of antiferromagnets in the Heisenberg model
• 4. Spin waves in bulk, thin films and stripes
• 5. Itinerant magnetism
• 6. Experimental techniques to study excitations

questions

Ferromagnetic resonance (FMR)

• 6. Experimental techniques to study excitations

Microwave absorption in a magnetic filed Often a constant frequency is used and the Larmor frequency is tuned into resonance by changing the applied field. Photon wavevector q=0, thus FMR detects mainly coherent precession modes.

Brillouin light scattering (BLS)

• 6. Experimental techniques to study excitations

Only q near the zone center accessible

Brillouin light scattering (BLS)

• 6. Experimental techniques to study excitations

Figure: B. Hillebrands

• 6. Experimental techniques to study excitations

Neutron Scattering Spallation neutron source in Oakridge, USA European neutron source in Grenoble

Inelastic neutron Scattering

• 6. Experimental techniques to study excitations

Inelastic (spin-flip) scattering of the neutron Access to full BZ but difficult near specular reflection (q=0). High energy resolution better than 1 meV possible. Only bulk samples due to weak interaction of neutrons with matter.

 Spin moment is transferred in the scattering process only by minority electrons.  Δq and ΔE of the scattered electron give q and E of the magnon.  Full access to the BZ but only at energies above a few 10 meV.

Spin-polarized Electron Energy Loss Spectroscopy (Sp-EELS)

• 6. Experimental techniques to study excitations

Sp-EELS of 8 ML fcc Co on Cu(100)

• M. Etzkorn, PhD thesis Universität Halle 2005
• 6. Experimental techniques to study excitations

• 0. Overview

Chapters of spin excitation

• 1. Why are excitations of any importance?
• 2. Excitations of ferromagnets in the Heisenberg model
• 3. Excitations of antiferromagnets in the Heisenberg model
• 4. Spin waves in bulk, thin films and stripes
• 5. Itinerant magnetism
• 6. Experimental techniques to study excitations

questions

Excitations of a ferromagnet Sz = -½ Sz = +½ magnon Sz = -1 Sz = -½ magnon Sz = -1 Sz = +½ Magnon creation Magnon annihilation

Inelastic tunneling spectroscopy

• Inelastic tunneling processes increase the number of

final states above a threshold.

• d²I/dU² is proportional to density of excitations if

elastic DOS is constant.

Inelastic tunnelling spectroscopy on paramagnets at 4K W tip on Cu(111) sample Clear spectrum, no features in DOS, no inelastic excitation channels

ITS on Fe(100) with W tip at 4K 1 mV mod. 150 nA 33% cross-section

Selection rules for magnon creation

 Excitation depends on direction of magnetic field  Prove of magnons and exclusion of phonons

B = - 7.5 mT B = + 7.5 mT Fe coated W tip on Fe(001) sample

Balashovet al., PRL 97, 182201 (2006)

Local excitation of magnons: Co/Cu(111)

 Peak intensity scales linearly with film thickness  Magnon creation cross-section: ~6% per ML

R = 2.05/2.85 = 0.72 ≈ ¾ 4 ML 3 ML Topography Mean free path λ ≈ 3 nm d2I/dU2 map

4 ML 3 ML 0 ML

Balashovet al., PRL 97, 182201 (2006)

ITS on Fe(100) with W tip at 4K 1 mV mod. 150 nA 33% cross-section Why does the magnon creation depend on the tunneling direction?

Selection rules for magnon creation in tunneling experiments Magnon creation occurs in the minority channel for positive bias Shows up as peak Magnon creation occurs in the majority channel for negative bias Shows up as dip non-magnetic tip magnetic sample

Forward-backward asymmetry of magnon creation A B

• 54%
• 61±3%

A B Fe(100)

Forward-backward asymmetry of magnon creation Co/Cu(111) 3 ML Co/Cu(100) 2 ML Pexp = -28±4% Pexp = -13±8% Ptheor = -26% Ptheor = -14% D O S DOS

Spin polarization and the spin torque Spin transfer due to magnon creation is proportional to the polarization of the tunneling current.

• Step in dI/dU is proportional to number of scattered electrons
• High efficiency of transfer of spin moment from current to magnetization

Quantum efficiency of the spin torque

Spin-torque effect

Spin-torque effect

M

Spin-flip scattering mechanism Magnon energies in Fe range from 0 to 600 meV Why are magnons created mainly near 10 meV by tunneling electrons?

Spin-flip scattering mechanism Hot electron Magnetization electron J (q) Born approximation

d  d  ∝∣J q ∣

2=ℑq

EF

Direct exchange between delocalized electrons In analogy to the exchange of localized electrons, the exchange for any two delocalized electrons is given by: E S−ET=2∫∫a

*r1b *r2

e

2

40∣r1−r2∣ ar2br1dr1 dr2 = 2 e

2

40 1 2

6∫∫∫

∣k1∣k F∫ ∣k 2∣k F

e

k1−k 2r1−r2

∣r1−r2∣

dr1dr2 dk1dk 2 (decomposition in Bloch waves inside Fermi sphere)

∫

∣k∣2k F e kr dk= 4

r ∫

2k F

k sinkrdk=−4  r

3 2k F cosk F r−sin2k F r

with follows E S−ET= −e

2

02

4∫

2k F cos2 K F r−sin2 K F r r

4

dr Oscillatory exchange: Rudeman-Kittel-Kasuya-Yoshida (RKKY) interaction Spin-flip scattering mechanism

Direct exchange between delocalized electrons In metals (e.g. Fe, Co, Ni) electrons are delocalized and form bands. Thus, exchange interaction extends beyond nearest neighbors.

Pajda et al. Phys. Rev. B 64, 17442 (2001)

Spin-flip scattering mechanism

RKKY: Spin-flip scattering mechanism Transfer of angular momentum between the tunneling current and the magnetic material reflects exchange interaction between delocalized electrons.

Balashov et al., PRB (2008)

J

• Series of magnon branches confined in the magnetic layer.
• One branch for every atomic layer.
• Quasi-momentum perpendicular to the film plane.

Magnons in thin films Quantized standing magnons

• Series of standing magnons confined in the magnetic layer.
• From energy, order and film thickness the dispersion relation can be obtained.

Magnon dispersion of fcc Co fcc Co/Cu(100)

d²I/dU²

9ML Co

• STM measurements agree well with neutron scattering data as well as with

ab initio theory.

• Spin wave stiffness of D=660 meVŲ is determined.

Magnon dispersion of fcc Co fcc Co/Cu(100)

Neutron scattering: Shirane et al., J. Appl. Phys. (1968) ab initio: Pajda et al., Phys. Rev. B (2001)

• Series of standing magnons confined in the antiferromagnetic layer.

Magnon dispersion of antiferromagnetic fcc Mn fcc Mn/Cu3Au(100) 5.8 ML 24 ML

• Dispersion nicely matches ab initio calculation and neutron scattering data.

Magnon dispersion of antiferromagnetic fcc Mn fcc Mn/Cu3Au(100) E= E gvsinka/a

2

Neutron scattering Eg=5.9 ± 0.2 meV v=185 ± 12 meVÅ STM measurements Eg=2.7 - 5.5 meV v=160 ± 10 meVÅ

Jankowska et al, JMMM,140, 1973 (1995)

• STM can measure magnon life times and surface anisotropies.

Magnon dispersion of antiferromagnetic fcc Mn Damping of spin waves Magnetic anisotropy energies

fcc Mn(17%Ni): Γ1= 85±18 meVÅ This work: Γ1= 39±8 meVÅ Exchange energy EE=25 meV Bulk anisotropy EAB=0.02±0.03 meV Surface anisotropy EAS=1.4±0.2 meV k=01 k

E g=2 E AE EE A

2 , E A=E ABE AS/t

C.L. Gao et al., PRL (2008)

Giant magnetic anisotropy

• P. Gambardella et al. Science 300 1130 (2003)

Uniaxial out-of-plane magnetic anisotropy of Co clusters on Pt(111) Giant magnetic anisotropy

• f 9.3 meV per Co atom

due to large orbital moment and thus large spin orbit interaction MAE drops quickly with cluster size Measurements by XMCD

• n ensemble of atoms
• r clusters

Large uncertainty on cluster sizes

Atomic manipulation Manipulation of the atoms by the STM tip

• Atoms are imaged with high tunneling resistance

(R>10MΩ) and moved with low tunneling resistances (R<300kΩ).

• Dimers and trimers appear higher than atoms.

17 Fe atoms on Cu(111) Forming a Co dimer on Pt(111)

Excitation due to tunneling electrons Spin-flip scattering of a tunneling electron

Sz = S Sz = S-1

−1/2 1/2

e- e-

The energy loss of the tunneling electron equals the magnetic excitation energy. D and Ku < 0 uniaxial anisotropy

A.J. Heinrich et al. Science 306 466 (2004)

ΔE cos(θ)=Sz/S Ku = ΔE * 2S²/(2S-1) DS2 E=DSz

2=Kucos(θ)

Inelastic tunneling spectroscopy Inelastic excitation energy E = 5.1meV for Fe/Pt(111) E = 10.2meV for Co/Pt(111)

• Phonon? E > 27meV
• Kondo effect? µ enhanced *
• Spin-flip

* T. Hermannsdörfer et al., J. Low Temp. 104 49 (1996)

What is the MAE of adatoms and clusters?

Balashov et al., PRL (2009)

What is the MAE of adatoms and clusters? Comparison with XMCD data

• P. Gambardella et al. Science 300 1130 (2003)

What is the life time of the excited state? Relaxation of the excited state Fe Co τ=55 fs τ=20 fs

• Life times can be extracted from excitation width,

corrected for instrumental broadening

High energy excitations of clusters Anisotropy and exchange

• For coherent rotation of all magnetic moments (rotation of total spin) the MAE has

to be overcome (S=3, Sz=3 → S=3 Sz=2). ΔE = -(3²-2²)D = -5D

• Also non collinear excitations are possible. In these cases also exchange energy

has to be paid (S=3, Sz=3 → S=2 Sz=2). ΔE = -5D+3J Fe2

• Fit to data gives J = 16+1 meV for Fe2
• Ab initio calculation of relaxed Fe dimer : J = 11 meV
• Fast relaxation of non-collinear state (10fs) via additional non spin flip process